Comencemos por eliminar esto: estoy atascado usando una base de datos de MS Access y no puedo cambiarlo.

Esto funciona bien:

using (OleDbConnection conn = ConnectionHelper.GetConnection())
{
  conn.Open();
  var results = conn.Query<string>(
    "select FirstName from Students where LastName = @lastName", 
    new { lastName= "Smith" }
  );
  conn.Close();
}

Esto funciona bien:

using (OleDbConnection conn = ConnectionHelper.GetConnection())
{
  OleDbCommand cmd = new OleDbCommand(
    "update Students set FirstName = @firstName, City = @city where LastName = @lastName", 
    conn
  );
  cmd.Parameters.AddWithValue("firstName", "John");
  cmd.Parameters.AddWithValue("city", "SomeCity");
  cmd.Parameters.AddWithValue("lastName", "Smith");

  conn.Open();
  var result = cmd.ExecuteNonQuery();
  conn.Close();
}

Esto no ... se ejecuta sin error, pero establece el Nombre como "SomeCity" en el DB y la Ciudad como "John":

using (OleDbConnection conn = ConnectionHelper.GetConnection())
{
  conn.Open();
  var results = conn.Query<string>(
    "update Students set FirstName = @firstName, City = @city where LastName = @lastName", 
    new { firstName = "John", city = "SomeCity", lastName = "Smith" }
  );
  conn.Close();
}

¿Algunas ideas?

EDITAR ABAJO

Dapper funciona si uso DynamicParameters:

using (OleDbConnection conn = ConnectionHelper.GetConnection())
{
  DynamicParameters parameters = new DynamicParameters();
  parameters.Add("firstName", "John");
  parameters.Add("city", "SomeCity");
  parameters.Add("lastName", "Smith");

  conn.Open();
  var result = conn.Query<string>(
    "update Students set FirstName = @firstName, City = @city where LastName = @lastName",
    parameters
  );
  conn.Close();
}